Group #27271415 — subgroups

Subgroups — #27271415

Order 13568.

244 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 63 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20273049	C₂	1	2	yes	maximal
2	C₅₃×#1818	C₂	1	1	yes	maximal, characteristic
3	#16810680	—	2	2	no	—
4	#16810349	C₄	1	4	yes	—
5	#16810347	—	2	2	no	—
6	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
7	C₂×C₂²⋊C₈×C₅₃	C₄	1	2	yes	—
8	C₅₃×#336	—	2	2	no	—
9	C₄⋊#88322	—	4	2	no	—
10	C₂²⋊#88322	—	2	4	no	—
11	#13133128	—	4	2	no	—
12	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₈×C₅₃	—	2	2	no	—
14	#117×C₅₃	—	4	2	no	—
15	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
16	C₂×C₂²⋊C₄×C₅₃	D₄	1	1	yes	characteristic
17	C₂×C₂²⋊C₄×C₅₃	Q₈	1	1	yes	characteristic
18	C₂×C₄²×C₅₃	D₄	1	2	yes	—
19	C₂²⋊C₈×C₅₃	—	4	2	no	—
20	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	—	2	2	no	—
22	C₂²×C₄×C₅₃	—	4	2	no	—
23	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
24	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
25	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
26	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
27	C₂×C₈×C₅₃	—	4	4	no	—
28	C₂×C₈×C₅₃	—	4	2	no	—
29	C₄⋊C₄×C₅₃	—	4	2	no	—
30	C₂²⋊C₄×C₅₃	—	4	1	no	—
31	C₂²⋊C₄×C₅₃	—	4	1	no	—
32	C₄²×C₅₃	—	2	2	no	—
33	C₄²×C₅₃	—	2	2	no	—
34	C₂×#88322	—	4	4	no	—
35	C₂³×C₅₃	—	2	1	no	—
36	C₂³×C₅₃	—	2	2	no	—
37	C₂³×C₅₃	—	4	2	no	—
38	C₂³×C₅₃	#95	1	1	yes	characteristic
39	C₂×C₄×C₅₃	—	2	2	no	—
40	C₂×C₄×C₅₃	—	2	1	no	—
41	C₂×C₄×C₅₃	—	2	2	no	—
42	C₂×C₄×C₅₃	—	2	2	no	—
43	C₂×C₄×C₅₃	—	2	2	no	—
44	C₂×C₄×C₅₃	—	4	4	no	—
45	C₂×C₄×C₅₃	—	4	2	no	—
46	C₂×C₄×C₅₃	—	4	2	no	—
47	C₂×C₄×C₅₃	#99	1	2	yes	—
48	C₈×C₅₃	—	8	2	no	—
49	#88322	—	8	4	no	—
50	#7327	—	53	1	no	maximal, Sylow
51	C₂²×C₅₃	—	2	2	no	—
52	C₂²×C₅₃	—	4	1	no	—
53	C₂²×C₅₃	—	4	2	no	—
54	C₂²×C₅₃	—	8	2	no	—
55	C₂²×C₅₃	#342	1	1	yes	characteristic
56	C₂²×C₅₃	#344	1	2	yes	—
57	C₄×C₅₃	—	2	2	no	—
58	C₄×C₅₃	—	4	1	no	—
59	C₄×C₅₃	—	4	2	no	—
60	C₄×C₅₃	—	4	2	no	—
61	#1818	D₅₃	1	1	yes	characteristic
62	#1464	—	53	2	no	—
63	C₂×C₅₃	—	2	2	no	—
64	C₂×C₅₃	—	8	1	no	—
65	C₂×C₅₃	#1306	1	1	yes	characteristic
66	C₂×C₅₃	#1395	1	2	yes	—
67	C₂×#117	C₂×D₅₃	1	1	yes	characteristic
68	#432	—	106	2	no	—
69	C₂×C₂²⋊C₈	Dic₅₃	1	2	yes	—
70	#336	—	2	2	no	—
71	#324	—	53	4	no	—
72	#323	—	106	2	no	—
73	C₅₃	#7327	1	1	yes	characteristic, complemented, Sylow
74	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
75	C₂²×C₈	—	2	2	no	—
76	#117	—	4	2	no	—
77	C₂×C₄⋊C₄	C₄×D₅₃	1	2	yes	—
78	C₂×C₂²⋊C₄	Dic₁₀₆	1	1	yes	characteristic
79	C₂×C₂²⋊C₄	D₂₁₂	1	1	yes	characteristic
80	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
81	C₄⋊C₈	—	212	2	no	—
82	C₂²⋊C₈	—	4	2	no	—
83	C₂²⋊C₈	—	106	4	no	—
84	#97	—	212	2	no	—
85	C₂⁴	C₄⋊Dic₅₃	1	1	yes	characteristic
86	C₂²×C₄	—	2	2	no	—
87	C₂²×C₄	—	4	2	no	—
88	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
89	C₂²×C₄	#11719919	1	2	yes	—
90	C₂²×C₄	#11719921	1	2	yes	—
91	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
92	C₂×C₈	—	4	4	no	—
93	C₂×C₈	—	4	2	no	—
94	C₂×C₈	—	212	4	no	—
95	C₄⋊C₄	—	4	2	no	—
96	C₂²⋊C₄	—	4	1	no	—
97	C₂²⋊C₄	—	4	1	no	—
98	C₄²	—	2	2	no	—
99	C₄²	—	2	2	no	—
100	C₂³	—	2	1	no	—
101	C₂³	—	2	2	no	—
102	C₂³	—	4	2	no	—
103	C₂³	#13133124	1	1	yes	characteristic
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	4	4	no	—
110	C₂×C₄	—	4	2	no	—
111	C₂×C₄	—	4	2	no	—
112	C₂×C₄	#13133132	1	2	yes	—
113	C₈	—	8	2	no	—
114	C₈	—	424	4	no	—
115	C₂²	—	2	2	no	—
116	C₂²	—	4	2	no	—
117	C₂²	—	4	1	no	—
118	C₂²	—	8	2	no	—
119	C₂²	#16810390	1	1	yes	characteristic
120	C₂²	#16810393	1	2	yes	—
121	C₄	—	2	2	no	—
122	C₄	—	4	2	no	—
123	C₄	—	4	2	no	—
124	C₄	—	4	1	no	—
125	C₂	—	2	2	no	—
126	C₂	—	8	1	no	—
127	C₂	#20272623	1	1	yes	characteristic
128	C₂	#20272844	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20273049

C₂

yes

maximal

C₅₃×#1818

C₂

yes

maximal, characteristic

#16810680

—

#16810349

C₄

yes

—

#16810347

—

C₂×#117×C₅₃

C₂²

yes

characteristic

C₂×C₂²⋊C₈×C₅₃

C₄

yes

—

C₅₃×#336

—

C₄⋊#88322

—

C₂²⋊#88322

—

#13133128

—

C₂³×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₃

—

#117×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₂×C₂²⋊C₄×C₅₃

D₄

yes

characteristic

C₂×C₂²⋊C₄×C₅₃

Q₈

yes

characteristic

C₂×C₄²×C₅₃

D₄

yes

—

C₂²⋊C₈×C₅₃

—

C₂⁴×C₅₃

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₄²

yes

characteristic

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₅₃

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₂×#88322

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

#95

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#99

yes

—

C₈×C₅₃

—

#88322

—

#7327

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#342

yes

characteristic

C₂²×C₅₃

#344

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

#1818

D₅₃

yes

characteristic

#1464

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#1306

yes

characteristic

C₂×C₅₃

#1395

yes

—

C₂×#117

C₂×D₅₃

yes

characteristic

#432

—

106

—

C₂×C₂²⋊C₈

Dic₅₃

yes

—

#336

—

#324

—

#323

—

106

—

C₅₃

#7327

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×Dic₅₃

yes

characteristic

C₂²×C₈

—

#117

—

C₂×C₄⋊C₄

C₄×D₅₃

yes

—

C₂×C₂²⋊C₄

Dic₁₀₆

yes

characteristic

C₂×C₂²⋊C₄

D₂₁₂

yes

characteristic

C₂×C₄²

C₂²⋊D₅₃

yes

—

C₄⋊C₈

—

212

—

C₂²⋊C₈

—

C₂²⋊C₈

—

106

—

#97

—

212

—

C₂⁴

C₄⋊Dic₅₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×Dic₅₃

yes

characteristic

C₂²×C₄

#11719919

yes

—

C₂²×C₄

#11719921

yes

—

C₂²×C₄

C₂²⋊Dic₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

212

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

100

C₂³

—

101

C₂³

—

102

C₂³

—

103

C₂³

#13133124

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

#13133132

yes

—

113

C₈

—

114

C₈

—

424

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

#16810390

yes

characteristic

120

C₂²

#16810393

yes

—

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₂

—

126

C₂

—

127

C₂

#20272623

yes

characteristic

128

C₂

#20272844

yes

—

	subgroup	G/H
center	C₂²	#16810390
commutator	C₂²×C₄×C₅₃	C₄²
Frattini	C₂×#117	C₂×D₅₃
Fitting	C₅₃×#1818	C₂
socle	C₂²×C₅₃	#342
prime radical	C₂⁴×C₅₃	C₄⋊C₄
cosocle	C₂×#117×C₅₃	C₂²
hypercenter	#1818	D₅₃
hypocenter	C₅₃	#7327
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ #1818

subgroup

G/H

center

C₂²

#16810390

commutator

C₂²×C₄×C₅₃

C₄²

Frattini

C₂×#117

C₂×D₅₃

Fitting

C₅₃×#1818

C₂

socle

C₂²×C₅₃

#342

prime radical

C₂⁴×C₅₃

C₄⋊C₄

cosocle

C₂×#117×C₅₃

C₂²

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ #1818

N	H ≅ G/N	type
73. C₅₃	#7327	semidirect

H ≅ G/N

type

73. C₅₃

#7327

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20273049	2
C₅₃×#1818	2
#7327	53

subgroup

index

#20273049

C₅₃×#1818

#7327

Subgroups — #27271415

Conjugacy classes of subgroups

Complements

Maximal subgroups